L11a265

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_22,12,9,11 X_18,8,19,7 X_20,18,21,17 X_16,22,17,21 X_2,9,3,10 X_4,16,5,15 X_14,6,15,5 X_6,14,7,13 X_8,20,1,19
Gauss code	{1, -7, 2, -8, 9, -10, 4, -11}, {7, -1, 3, -2, 10, -9, 8, -6, 5, -4, 11, -5, 6, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)+t(2)-1)(t(2)t(1)-t(1)-t(2))(2t(2)t(1)-t(1)-t(2)+2)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$17q^{9/2}-16q^{7/2}+11q^{5/2}-7q^{3/2}+q^{21/2}-3q^{19/2}+6q^{17/2}-11q^{15/2}+15q^{13/2}-17q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-3}-2z^{5}a^{-5}-z^{5}a^{-7}+z^{3}a^{-1}-3z^{3}a^{-5}-z^{3}a^{-7}+z^{3}a^{-9}+za^{-1}+2za^{-3}-za^{-5}-za^{-7}+za^{-9}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-3z^{9}a^{-5}-6z^{9}a^{-7}-3z^{9}a^{-9}-5z^{8}a^{-4}-8z^{8}a^{-6}-7z^{8}a^{-8}-4z^{8}a^{-10}-5z^{7}a^{-3}-5z^{7}a^{-5}+4z^{7}a^{-7}+z^{7}a^{-9}-3z^{7}a^{-11}-3z^{6}a^{-2}+4z^{6}a^{-4}+14z^{6}a^{-6}+17z^{6}a^{-8}+9z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}+8z^{5}a^{-3}+18z^{5}a^{-5}+10z^{5}a^{-7}+10z^{5}a^{-9}+9z^{5}a^{-11}+5z^{4}a^{-2}+2z^{4}a^{-4}-6z^{4}a^{-6}-11z^{4}a^{-8}-5z^{4}a^{-10}+3z^{4}a^{-12}+2z^{3}a^{-1}-6z^{3}a^{-3}-18z^{3}a^{-5}-13z^{3}a^{-7}-11z^{3}a^{-9}-8z^{3}a^{-11}-2z^{2}a^{-2}-3z^{2}a^{-4}+4z^{2}a^{-8}+z^{2}a^{-10}-2z^{2}a^{-12}-za^{-1}+4za^{-3}+7za^{-5}+4za^{-7}+4za^{-9}+2za^{-11}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

22

1

-1

20

2

18

4

1

-3

16

7

2

5

14

8

4

-4

12

9

7

2

10

9

0

8

7

8

-1

6

4

9

5

4

3

7

-4

2

1

5

4

0

2

-2

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=2$	$i=4$
$r=-2$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=9$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$